Start with the equation:

\(4 \sin \theta + \tan \theta = 0\)

Substitute \(\tan \theta = \frac{\sin \theta}{\cos \theta}\):

\(4 \sin \theta + \frac{\sin \theta}{\cos \theta} = 0\)

Multiply through by \(\cos \theta\) to eliminate the fraction:

\(4 \sin \theta \cos \theta + \sin \theta = 0\)

Factor out \(\sin \theta\):

\(\sin \theta (4 \cos \theta + 1) = 0\)

This gives two cases:

1. \(\sin \theta = 0\)

2. \(4 \cos \theta + 1 = 0\)

For \(\sin \theta = 0\), \(\theta = 0^\circ, 180^\circ\), but these are outside the given range.

For \(4 \cos \theta + 1 = 0\):

\(\cos \theta = -\frac{1}{4}\)

Using the inverse cosine function, \(\theta \approx 104.5^\circ\) within the range \(0^\circ < \theta < 180^\circ\).